The Limits of Knowledge

Posted on 16 August 2005

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Mathematics is (from one point of view anyway) the study of a particularly rigorous and precise kind of knowledge. Although it is quite specialized, it nonetheless can open us up to insights into the nature of knowledge in general. One example of this comes from the mathematician Kurt Godel, who proved two remarkable theorems that shook the foundations of mathematics. If we reflect on these theorems, they have the potential to shake the foundations of our own knowledge as well.

Before Godel proved his two theorems in 1931, a number of eminent mathematicians were engaged in a program whose goal was to objectify all of mathematics, i.e., make it into a complete and consistent system of symbols and rules for manipulating them. One motivation for this program was to eliminate inconsistencies and paradoxes in mathematics that might be hidden in vague intuitions and informal arguments. The idea was that if you could make all your assumptions and rules of reasoning completely explicit in a system of symbols and rules for manipulating those symbols, then there would be no room for inconsistencies to creep in. The challenge was then to find and make explicit a set of axioms and rules of logical inference that would provide a foundation for all of mathematics. If such a goal were realized, using the rules of logical inference we would be able to deduce every truth of mathematics from the axioms by manipulating symbols in a completely explicit and mechanical way, without appeal to any intuitions or informal arguments. Mathematical knowledge would be completely objective.

Godel pulled the rug out from under this program and showed that it is impossible to objectify all of mathematical knowledge in this way. Assuming that a set of axioms is consistent and sufficiently complex to define basic arithmetic, Godel proved that there are mathematical truths that can not be proved within the system itself. The system, in other words, is incomplete. It can not encompass all truths. Even if one were to expand the system by adding another axiom or two, Godel’s theorem says that there would still be another truth that is beyond the power of this expanded system to prove. Thus, it is impossible for mathematical knowledge to be completely objectified. Or, to put it another way, there will always be a dimension of mathematical knowledge that transcends any objective system of mathematics. There will never be any final mathematical system, any complete and explicit description of mathematical truth. Like an eternal spring, mathematics flows from an inexhaustible, hidden source.

Like the program to objectify all of mathematics in one system, we sometimes think that we can understand all of reality with our logical systems of thought. Whether it be a grand unified theory of physics or a master philosophical system, there is a tendancy to think that reality can be defined and captured by systems of thought. Or we slip into the opposite extreme and think that reality can not be known at all. What Godel’s theorem suggests, however, is another possibility. Rather than viewing our quest for deeper knowledge as an ultimately futile project whose goal is to completely capture reality in an objective system of thought, we can see it as an invitation to participate in deeper and more subtle communion that infinite source from which all objective knowledge arises. Although objective systems of thought can comprehend only a limited part of reality, our knowledge is not limited to objective systems of thought.

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Posted in: Philosophy, Science